Zero-free regions

The table below lists various regions of the [math](t,y,x)[/math] parameter space where [math]H_t(x+iy)[/math] is known to be non-zero. In some cases the parameter

[math] N := \lfloor \sqrt{\frac{x}{4\pi} + \frac{t}{16}} \rfloor[/math]

is used instead of [math]x[/math]. The mesh evaluation techniques also require rigorous upper bounds on derivatives. In some cases the spacing of the mesh is fixed; in other cases it is adaptive based on the current value of the evaluation and on the derivative bound.

Date	[math]t[/math]	[math]y[/math]	[math]x[/math]	From	Method	Comments
1950	[math]t \geq 0[/math]	[math]y \gt \sqrt{\max(1-2t,0)}[/math]	Any	De Bruijn	Theorem 13 of de Bruijn	Proves [math]\Lambda \leq 1/2[/math].
2004	0	[math]y\gt0[/math]	[math]0 \leq x \leq 4.95 \times 10^{11}[/math]	Gourdon-Demichel	Numerical verification of RH & Riemann-von Mangoldt formula	Results have not been independently verified
2009	[math]t \gt 0[/math]	[math]y \gt 0[/math]	[math]x \geq C(t)[/math]	Ki-Kim-Lee	Theorem 1.3 of Ki-Kim-Lee	[math]C(t)[/math] is not given explicitly. Also they show [math]\Lambda \lt 1/2[/math].
2017	0	[math]y\gt0[/math]	[math]0 \leq x \leq 6.1 \times 10^{10}[/math]	Platt	Numerical verification of the Riemann hypothesis
Mar 7 2018	0.4	0.4	[math]N \geq 2000[/math] ([math]x \geq 5.03 \times 10^7[/math])	Tao	Analytic lower bounds on [math]A^{eff}+B^{eff} / B^{eff}_0[/math] and analytic upper bounds on error terms	Can be extended to the range [math]0.4 \leq y \leq 0.45[/math]
Mar 10 2018	0.4	0.4	[math]151 \leq N \leq 300[/math] ([math]2.87 \times 10^5 \leq x \leq 1.13 \times 10^6[/math])	KM	Mesh evaluation of [math]A^{eff}+B^{eff} / B^{eff}_0[/math] and upper bounds on error terms
Mar 11 2018	0.4	0.4	[math]300 \leq N \leq 2000[/math] ([math]1.13 \times 10^6 \leq x \leq 5.03 \times 10^7[/math])	KM	Analytic lower bounds on [math]A^{eff}+B^{eff} / B^{eff}_0[/math] and upper bounds on error terms	Should extend to the range [math]0.4 \leq y \leq 0.45[/math]
Mar 11 2018	0.4	0.4	[math]20 \leq N \leq 150[/math] ([math]5026 \leq x \leq 2.87 \times 10^5[/math])	Rudolph & KM	Mesh evaluation of [math]A^{eff}+B^{eff} / B^{eff}_0[/math] and upper bounds on error terms
Mar 11 2018	0.4	0.4	[math]11 \leq N \leq 19[/math] ([math]1520 \leq x \leq 5026[/math])	Rudolph & KM	Mesh evaluation of [math]A^{eff}+B^{eff} / B^{eff}_0[/math] and upper bounds on error terms
Mar 22 2018	0.4	0.4	[math]x \leq 1000[/math]	Anon/David/KM	Mesh evaluation of [math]H_t[/math]
Mar 22 2018	0.4	0.4	[math]1000 \leq x \leq 1600[/math]	Rudolph	Mesh evaluation of [math]H_t[/math]
Mar 22 2018	0.4	0.4	[math]8 \leq N \leq 10[/math] ([math]803 \leq x \leq 1520[/math])	Rudolph	Mesh evaluation of [math]A^{eff}+B^{eff} / B^{eff}_0[/math] and upper bounds on error terms
Mar 23 2018	0.4	0.4	[math]20 \leq x \leq 1000[/math]	Anonymous	Mesh evaluation of [math]H_t[/math]
Mar 23 2018	[math]t \gt 0[/math]	[math]y \gt 0[/math]	[math]x \gt \exp(C/t)[/math]	Tao	Analytic bounds on [math]A^{eff}+B^{eff} / B^{eff}_0[/math] and error terms; argument principle	[math]C[/math] is in principle an explicit absolute constant
Mar 27 2018	0.4	[math]0.4 \leq y \leq 0.45[/math]	[math]7 \leq N \leq 300[/math] ([math]615 \leq x \leq 1.13 \times 10^6[/math])	KM	Mesh evaluation of [math]A^{eff}+B^{eff} / B^{eff}_0[/math] and upper bounds on error terms; argument principle
Mar 27 2018	0.4	[math]0.4 \leq y \leq 0.45[/math]	[math]0 \leq x \leq 1000[/math]	Anonymous	Mesh evaluation of [math]H_t[/math]; argument principle	Completes proof of [math]\Lambda \leq 0.48[/math]!
Mar 31 2018	[math]0 \leq t \leq 0.4[/math]	[math]0.4 \leq y \leq 1[/math]	[math]10^6 \leq x \leq 10^6 + 1 [/math]	KM	Mesh evaluation of [math]A^{eff}+B^{eff} / B^{eff}_0[/math] and upper bounds on error terms; argument principle
Mar 31 2018	0.4	[math]0.4 \leq y \leq 0.45[/math]	[math]0 \leq x \leq 3000[/math]	Rudolph	Third approach to [math]H_t[/math]; argument principle
Apr 6 2018	[math]0 \leq t \leq 0.2[/math]	[math]0.4 \leq y \leq 1[/math]	[math]5 \times 10^9 \leq x \leq 5 \times 10^9+1[/math]	KM, Rudolph, David, Anonymous	Mesh evaluation of [math]A^{eff}+B^{eff} / B^{eff}_0[/math] and upper bounds on error terms; argument principle
Apr 6 2018	0.2	0.4	[math]N \geq 3 \times 10^6 (x \geq 1.13 \times 10^{12})[/math]	KM	Analytic lower bounds on [math]A^{eff}+B^{eff} / B^{eff}_0[/math] and upper bounds on error terms
Apr 7 2018	0.29	[math]y \geq 0.29[/math]	[math]N \geq 19947[/math]	Anonymous	Triangle inequality bound on [math]A^{eff}+B^{eff} / B^{eff}_0[/math] and upper bounds on error terms	Would in principle show [math]\Lambda \leq 0.33205[/math] if the matching barrier could be established
Apr 9 2018	0.2	[math]y \geq 0.4[/math]	[math]N \geq 3 \times 10^5[/math]	Tao	Triangle inequality bound on [math]A^{eff}+B^{eff} / B^{eff}_0[/math] and upper bounds on error terms
Apr 10 2018	0.2	[math]y \geq 0.4[/math]	[math]4 \times 10^4 \leq N \leq 10^5; 100|N[/math]	KM	Euler2 mollifier and triangle inequality bounds on [math]A^{eff}+B^{eff} / B^{eff}_0[/math]	Error terms not estimated but look well within acceptable limits
Apr 12 2018	0.2	[math]y \geq 0.4[/math]	[math]4 \times 10^4 \leq N \leq 3 \times 10^5[/math]	Anonymous	Euler2 mollifier and triangle inequality bounds on [math]A^{eff}+B^{eff} / B^{eff}_0[/math]	Error terms not estimated but look well within acceptable limits
Apr 12 2018	0.2	[math]y \geq 0.4[/math]	[math]19947 \leq N \leq 4 \times 10^4[/math]	Rudolph	Euler3 mollifier and triangle inequality bounds on [math]A^{eff}+B^{eff} / B^{eff}_0[/math]	Error terms not estimated but look well within acceptable limits